Asymptotics of spectrum of multipoint differential operators with summable potential

S. I. Mitrokhin

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 17 (2017) no. 2, pp. 69-81

Résumé

We study the boundary value problem for a differential operator with summable potential and study the asymptotic behavior of solutions for large values of the spectral parameter.

Détail

Keywords: differential operator, boundary value problem, summable potential, multipoint boundary conditions, asymptotics of the eigenvalues.

Bibliographie

[1] M. A. Naimark, Linear Differential Operators, Nauka, M., 1969 (in Russian)

[2] V. B. Lidsky, V. A. Sadovnichiy, “Regularized sums of the roots of one class of the entire functions”, Funkz. Analyz i ego Prilozh., 1:2 (1967), 52–59 (in Russian)

[3] M. G. Gasymov, B. M. Levitan, “Determination of differential operators from two spectrums”, Uspekhi Mat. Nauk, 19:2 (1964), 3–63 (in Russian)

[4] Gelfand I. M., Levitan B. M., “About one simple identity for eigenvalues of differential operator of the second order”, Dokl. AN SSSR, 88 (1953), 593–596 (in Russian) | Zbl

[5] M. V. Keldysh, “About eigenvalues and eigenfunctions of certain classes of nonselfadjoint equations”, Dokl. AN SSSR, 77 (1951), 11–14 (in Russian) | Zbl

[6] V. A. Ilyin, “On the convergence of expansions on eigenfunctions at points of the discontinuity of the coefficients of the differential operator”, Mat. Zametki, 22:5 (1977), 698–723 (in Russian)

[7] S. I. Mitrokhin, “On the formulas of regularized traces for differential operators of the second order with discontinuous coefficients”, Vestn. MGU. Ser. Mat, Mekh., 1986, no. 6, 3–6 (in Russian)

[8] V. D. Budaev, “About an unconditional basis property on a closed interval of systems of eigen and adjointed functions of a second order operator with discontinuous coefficients”, Differ. Uravn., 23:6 (1987), 941–952

[9] S. I. Mitrokhin, “On the trace formulas for a boundary value problem with a functional-differential equation with a discontinuous coefficient”, Differ. Uravn., 22:6 (1986), 927–931 (in Russian) | Zbl

[10] V. A. Ilyin, “On the Riesz basis property of systems of root vector functions of the discontinuous operator of Schredinger with a matrix potential”, Dokl. AN SSSR, 314:11 (1990), 59–62 (in Russian) | Zbl

[11] S. I. Mitrokhin, “On certain spectral properties of the differential operators of the second order with discontinuous weight function”, Dokl. Ross. Akad. Nauk, 356:1 (1997), 13–15 (in Russian)

[12] S. I. Mitrokhin, “On the “splitting” of multiples in the main eigenvalues of multipoint boundary value problems”, Izv. Vyssh. Uchebn. Zaved. Mat., 1997, no. 3(418), 38–43 (in Russian)

[13] V. A. Vinokurov, V. A. Sadovnichiy, “The asymptotics of any order of the eigenvalues and eigenfunctions of the boundary value problem of Sturm–Liouville on an interval with summable potential”, Differ. Uravn., 34:10 (1998), 1423–1426 (in Russian) | Zbl

[14] V. A. Vinokurov, V. A. Sadovnichiy, “The asymptotics of any order of the eigenvalues and eigenfunctions of the boundary value problem of Sturm—Liouville on an interval with summable potential”, Izv. Akad. Nauk SSSR Ser. Mat., 64:4 (2000), 47–108 (in Russian) | DOI | Zbl

[15] S. I. Mitrokhin, “On the spectral properties of a fourth-order differential operator with summable coefficients”, Proc. of Steklov Math. Inst., 270, 2010, 188–197 (in Russian) | Zbl

[16] S. I. Mitrokhin, “Spectral properties of boundary value problems for functional differential equations with integrable coefficients”, Differ. Uravn., 46:8 (2010), 1085–1093 (in Russian) | Zbl

[17] S. I. Mitrokhin, “On the spectral properties of a differential operator with a summable coefficients with a retarded argument”, Ufa Math. Zh., 3:4 (2011), 95–115 (in Russian)

[18] S. I. Mitrokhin, “Asymptotics of the eigenvalues of the differential operator of the tenth order with summable potential”, Usp. Sovr. Estv., 2010, no. 3, 146–149 (in Russian)

[19] S. I. Mitrokhin, “About spectral properties of differential operators of odd order with summable potential”, Differ. Uravn., 47:12 (2011), 1808–1811 (in Russian) | Zbl

[20] B. M. Levitan, I. S. Sargsyan, Introduction to the Spectral Theory, Nauka, M., 1970

[21] V. A. Yurko, Introduction to the Theory of Inverse Spectral Problems, Fizmatlit, M., 2007 (in Russian)

[22] R. Bellman, K. L. Cooke, Differential-Difference Equations, Mir, M., 1967 (in Russian)

[23] V. A. Sadovnichiy, V. A. Lyubishkin, “About some new results of the theory of regularized traces of differential operators”, Differ. Uravn., 18:1 (1982), 109–116 (in Russian)

[24] Sadovnichiy V. A., Lyubishkin V. A., Belabassi Yu., “On regularized sums of the roots of an entire functions of one class”, Dokl. AN SSSR, 254:6 (1980), 1346–1348 (in Russian) | Zbl